Algebraic structures of F-manifolds via pre-Lie algebras

Abstract

We relate the operad FMan controlling the algebraic structure on the tangent sheaf of an F -manifold (weak Frobenius manifold) defined by Hertling and Manin to the operad PreLie of pre-Lie algebras: for the filtration of PreLie by powers of the ideal generated by the Lie bracket, the associated graded object is FMan.

Three Schur functors related to pre-Lie algebras

“…Moreover, Bolgar proved in [Bol96] that for the natural filtration on U PreLie (g), the associated graded pre-Lie algebra is isomorphic to the universal enveloping pre-Lie algebra of the abelian Lie algebra with the same underlying vector space as g, which brings universal enveloping pre-Lie algebras closer to the context of the category theoretical approach to PBW theorems proposed by the first author and Tamaroff in [DT18] who proved that a functorial PBW theorem holds in this case using a previous result [Dot19] of the first author. Neither of the abovementioned results, however, leads to an explicit description of the Schur functor of U PreLie (g).…”

“…Our result has some immediate applications, of which we give three: a proof of a similar result for the operad of F -manifold algebras [Dot19], a formula for the permutative bar homology of an associative commutative algebra, and a hint that can hopefully be used to construct a conjectural good triple of operads [Lod08] (X c , PreLie, Lie) that would allow one to prove a Milnor-Moore theorem for universal enveloping pre-Lie algebras.…”

On F-algebroids and Dubrovin’s duality

“…We want to thank Yuri Manin for reading a draft of this note and his valuable comments. In particular, he drew our attention to the paper [4] by V. Dotsenko and pointed out the possibility of finding relations between the approach in that paper and ours. A. Torres-Gomez is partially supported by Universidad del Norte grant number 2018-17 "Agenda I+D+I".…”

“…Finally, we think that giving an algebraic definition of our F-algebroid (similar to the Lie-Rinehart pair description of a Lie algebroid) we could make contact with the construction of the operad FMan in [4], this is work in progress.…”

F-manifold color algebras

“…In 1999, Hertling and Manin [12] introduced the concept of F-manifolds as a relaxation of the conditions of Frobenius manifolds. Inspired by the investigation of algebraic structures of F-manifolds, the notion of an F-manifold algebra is given by Dotsenko [8] in 2019 to relate the operad F-manifold algebras to the operad pre-Lie algebras. An F-manifold algebra is defined as a triple (A, •, [, ]) satisfying the following Hertling-Manin relation, P x•y (z, w) = x • P y (z, w) + y • P x (z, w), ∀x, y, z, w ∈ A, where (A, •) is a commutative associative algebra, (A, [, ]) is a Lie algebra and…”